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hexagon:2610.00021v1

math.MG — Metric Geometry

A Fourier Consequence for Lattice Kissing Numbers and Average Contacts

Contributed by Scott Kominers

We deduce a new asymptotic upper bound Klat(n)≤(2e/π+o(1))nK_{\mathrm{lat}}(n)\leq\left(\sqrt{2e/\pi}+o(1)\right)^n on the maximal lattice kissing number Klat(n)K_{\mathrm{lat}}(n) in dimension nn, using the explicit auxiliary functions constructed in OpenAI's "Ten Advances" preprint. The same bound holds for the average contact degree of a finite packing of congruent balls. The bound's base-22 exponential rate rounds to 0.39560.3956, matching the value extrapolated empirically by Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini in 2020.

A mix of human-written and AI-generated textHuman understanding: all partsContact numbersDiscrete geometryEuclidean latticesFourier linear programming boundsGeometry of numbersKissing numbersSphere packing
Oct 02, 2026PDF ↗View work →
hexagon:2609.00123v1

math.MG — Metric Geometry

Breaking the Fourth Wall: The Planar Centroid Banach–Mazur Diameter Is Strictly Less than 44

Contributed by Scott Kominers

We prove that the centroid Banach–Mazur diameter of planar convex bodies is strictly less than 44, (slightly) improving the prior upper bound 69/17≈4.0588269/17\approx 4.05882 due to Lassak. The proof combines a sharp covariance-ellipse sandwich, whose equality cases in any single direction occur only for triangles, with a compactness argument: an extremal pair at distance 44 would consist of two triangles, which are linearly equivalent and hence must in fact be at distance 11. The resulting gap below 44 is uniform but nonquantitative.

A mix of human-written and AI-generated textHuman understanding: all partsBanach–Mazur diameter
Sep 28, 2026PDF ↗View work →